Phase-field modeling of fracture in heterogeneous materials: jump conditions, convergence and crack propagation
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O R I G I NA L
Arne Claus Hansen-Dörr Kästner
· Jörg Brummund · Markus
Phase-field modeling of fracture in heterogeneous materials: jump conditions, convergence and crack propagation
Received: 3 April 2020 / Accepted: 19 August 2020 © The Author(s) 2020
Abstract In this contribution, a variational diffuse modeling framework for cracks in heterogeneous media is presented. A static order parameter smoothly bridges the discontinuity at material interfaces, while an evolving phase-field captures the regularized crack. The key novelty is the combination of a strain energy split with a partial rank-I relaxation in the vicinity of the diffuse interface. The former is necessary to account for physically meaningful crack kinematics like crack closure, the latter ensures the mechanical jump conditions throughout the diffuse region. The model is verified by a convergence study, where a circular bi-material disc with and without a crack is subjected to radial loads. For the uncracked case, analytical solutions are taken as reference. In a second step, the model is applied to crack propagation, where a meaningful influence on crack branching is observed, that underlines the necessity of a reasonable homogenization scheme. The presented model is particularly relevant for the combination of any variational strain energy split in the fracture phase-field model with a diffuse modeling approach for material heterogeneities. Keywords Phase-field modeling · Diffuse modeling framework · Incremental variational formulation · Mechanical jump conditions
1 Introduction Modern engineering simulation challenges comprise the prediction of failure, which is one of the most severe mechanism affecting the bearing capacity. The phase-field method for the simulation of crack growth proved to be a powerful tool because it incorporates crack nucleation and arrest, as well as branching and merging of cracks [1–3]. Based on the variational approach to brittle fracture [4], it regularizes the underlying energy functional [5] and approximates the crack by an auxiliary scalar field, which is referred to as crack phasefield. The phase-field smoothly bridges the intact and fully broken state by introducing a length scale c . The approach is consistent with the energetic cracking criterion introduced by Griffith [6]. The phase-field method for crack modeling allows for fixed meshes where the element edges do not have to be aligned with the crack path, i.e., remeshing is avoided when the crack changes its direction or branches. The approach for a non-conforming description can be extended to a more general setting, where all discontinuities within a heterogeneous structure are captured in such a way. Cumbersome preprocessing such as manual identification of heterogeneities and meshing can be skipped or at least simplified if the structure is for example obtained from direct imaging [7–9]. Prominent representatives are the extended finite element method [10–12] or the finite cell method [13,14]. The latter is among the category of fictitious domain met
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